Bahar FataDepartment of Bioengineering,

University of Pittsburgh,

Pittsburgh, PA 15219

Danielle Gottlieb

John E. Mayer

Cardiac Surgery Program,

Boston Children’s Hospital,

Harvard Medical School,

Boston, MA 02115

Michael S. Sacks1

Professor of Biomedical Engineering

Department of Biomedical Engineering,

Institute for Computational Engineering

and Science,

University of Texas,

Austin, TX 78712

e-mail: msacks@ices.utexas.edu

Estimated in Vivo PostnatalSurface Growth Patternsof the Ovine Main PulmonaryArtery and Ascending AortaDelineating the normal postnatal development of the pulmonary artery (PA) and ascend-ing aorta (AA) can inform our understanding of congenital abnormalities, as well as pul-monary and systolic hypertension. We thus conducted the following study to delineate thePA and AA postnatal growth deformation characteristics in an ovine model. MR imageswere obtained from endoluminal surfaces of 11 animals whose ages ranged from 1.5months/15.3 kg mass (very young) to 12 months/56.6 kg mass (adult). A bicubic Hermitefinite element surface representation was developed for the each artery from each animal.Under the assumption that the relative locations of surface points were retained duringgrowth, the individual animal surface fits were subsequently used to develop a method toestimate the time-evolving local effective surface growth (relative to the youngest meas-ured animal) in the end-diastolic state. Results indicated that the spatial and temporalsurface growth deformation patterns of both arteries, especially in the circumferentialdirection, were heterogeneous, leading to an increase in taper and increase in cross-sectional ellipticity of the PA. The longitudinal PA growth stretch of a large segment onthe posterior wall reached 2.57 6 0.078 (mean 6 SD) at the adult stage. In contrast, thelongitudinal growth of the AA was smaller and more uniform (1.80 6 0.047). Interest-ingly, a region of the medial wall of both arteries where both arteries are in contactshowed smaller circumferential growth stretches—specifically 1.12 6 0.012 in the PAand 1.43 6 0.071 in the AA at the adult stage. Overall, our results indicated that contactbetween the PA and AA resulted in increasing spatial heterogeneity in postnatal growth,with the PA demonstrating the greatest changes. Parametric studies using simplified geo-metric models of curved arteries during growth suggest that heterogeneous effective sur-face growth deformations must occur to account for the changes in measured arterialshapes during the postnatal growth period. This result suggests that these first results area reasonable first-approximation to the actual effective growth patterns. Moreover, thisstudy clearly underscores how functional growth of the PA and AA during postnatal mat-uration involves complex, local adaptations in tissue formation. Moreover, the presentresults will help to lay the basis for functional replacement by defining critical geometricmetrics. [DOI: 10.1115/1.4024619]

Introduction

The normal development of the pulmonary artery (PA) can begreatly affected in many congenital heart defects, often leading topulmonary hypertension if left untreated [1,2]. Congenital abnor-malities of these arteries often necessitate surgical repair or theuse of a valved conduit replacement [3–5], requiring multiplereinterventions due to regurgitation or failure of the prostheticconduit [6]. In recent years, there has been a growing interest inthe development of a living autologous tissue graft that couldaddress the critical need for growing substitutes in the repair ofcongenital cardiovascular defects [7–9]. Regardless of the particu-lars of the therapeutic approach, the detailed growth characteris-tics of the native artery are required to establish the baselinedimensional changes post-implantation. Moreover, delineating thenormal arterial growth patterns has implications for the timingand nature of surgical repair [3,10]. The geometries of the PA andascending aorta (AA) are also important indicators of pulmonaryand systolic hypertension [11], such as the ratio of diameters of

the PA to the AA is an important measure of pulmonary hyperten-sion in younger adults [12].

During the prenatal period, a state of physiologic pulmonaryhypertension exists due to the equalization of pressures by the pat-ent ductus arteriosus, resulting in similar wall thickness of the AAand the PA. After birth, as the ductus arteriosus closes, the pulmo-nary arterial pressure decreases. After the first year of life, thick-ness of the PA is normally less than half that of the adjacentascending aorta, although the diameters of the two great arteriesremain the same relative to one another [13]. During homeostaticconditions, the total pulmonary and systemic blood flows areessentially identical. In spite of their comparable blood flow rate,the anatomic characteristics of these two segments of the cardio-vascular system differ substantially [14].

The significance of postnatal remodeling in the geometry of thegreat arteries (mainly for the aorta) has been previously identifiedand basic dimensional changes during the postnatal growth periodreported [15,16]. An understanding of the dynamic processes ofdevelopmental growth in arteries requires a detailed description ofthe temporal and spatial patterns of geometrical changes. How-ever, the full postnatal surface growth deformation of thesearteries has yet to be established. Moreover, despite their closemechanical association throughout life, the effects of the AA andPA physical interactions on their respective geometries during the

1Corresponding author.Contributed by the Bioengineering Division of ASME for publication in the

JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received December 26, 2012;final manuscript received April 26, 2013; accepted manuscript posted May 22, 2013;published online June 11, 2013. Assoc. Editor: Keith Gooch.

Journal of Biomechanical Engineering JULY 2013, Vol. 135 / 071010-1Copyright VC 2013 by ASME

normal postnatal remodeling period remain unknown. In a recentstudy using basic geometric characterization approach, we demon-strated that the PA did not undergo a strictly affine scaling duringnormal postnatal growth [17], but rather demonstrated heterogene-ous growth patterns. Specifically, it was shown that the PAbecomes increasingly tapered with maturation along with anincrease in the cross-sectional ellipticity, with quite differentgrowth patterns in the AA. These preliminary results suggestedthat during postnatal development each artery is possibly guidedby local physical interactions resulting in heterogeneous growthpatterns.

These considerations point towards a need for a basic under-standing of the driving process of the growth trends in the PA andAA. In the kinematics of finite growth, the effects of growth arerepresented by the transformation of the stress-free state using amultiplicative decomposition of the total deformation gradient Finto an elastic Fe and a growth Fg component, so that F¼FeFg

[18]. However, no such information currently exists for the post-natal development of the PA and AA. Moreover, the undertakingof detailed investigations to determine Fg and Fe need to be pre-ceded by studies to establish basic patterns to see if such studiesare fully warranted and to guide the experimental designs. Ourgoal was to shed light on the specific effective growth deforma-tion patterns in these arteries during postnatal development.

Methods

Imaging, Segmentation, and Registration. The methodologyfor obtaining the 3D in vivo arterial geometry data using an ovinemodel has been previously presented [17]. Briefly, the in vivo end-diastolic magnetic resonance images (MRI) of both arteries’ endo-luminal surfaces in eleven different animals (Ovis aries, subspeciesDorset) weighing between 15.3 to 56.6 kg, of ages 1.5 (the earliesttime point used for ovine heart valve implants [19]) to 12 months(full adulthood), were obtained. Imaging was performed with a 1.5T MRI scanner (Philips Achieva, Best, the Netherlands) and a5-channel cardiac radio frequency surface coil. After localizingimages, breath-hold ECG-gated steady-state free precessionsequences of thorax were obtained in long and short-axis ventricu-

lar planes, and in the long and short-axis planes to the pulmonaryvalve. A three-dimensional isotropic data set (with resolution of1.5� 1.5� 1.5 mm sections, reconstructed to 0.78� 0.78� 0.78mm) was acquired at end-expiration and end-diastole. The segmen-tation masks of MR images of the PA and AA endoluminal surfaceswere acquired using the semiautomated shells and spheres method[20]. The algorithm also generated a set of paired boundary andlocal center locations [20]. The local center points helped define thecenterline paths used in registering PAs as well as to develop anappropriate curvilinear coordinate system to carry out the deforma-tion analysis. In the present work, the PA was defined from the pul-monary sinotubular junction (STJ) to the pulmonary bifurcationpoint (BFN), and the AA from the aortic STJ to the BFN-intersecting plane of the AA. The section of each artery so definedwas utilized as the region of interest (ROI) (Fig. 1).

While semiautomated nonrigid 3D registration methods thatcan deform one image to match the template image by defining aspatial transformation exist [21], the inherent variations of the PAand AA surface geometries precluded their use. The AA and PAdatasets were therefore registered separately through the follow-ing global affine transformation sequence. The affine registrationof the PA was defined from three significant features: the center-line path, and the left and right pulmonary arteries at the BFN(Fig. 1(a)). The AA registration was based on the main sidebranch and the two coronary arteries at the sinuses of Valsalva(Fig. 1(a)), which formed three prominent anatomical landmarks.The AA and PA of the youngest animal (1.5 month-old, 15.3 kg inmass) were chosen as the referential templates to define registra-tion. Each segmentation image was aligned to the correspondingtemplate using the following two-step registration procedure. Anautomated intensity-based registration was first used to align theimages, and then the images were manually transformed toimprove alignment of the aforementioned features. The automatedregistration was performed within a standard registration frame-work consisting of a cost function, optimizer, transform, and inter-polator. In this framework, a cost function was computed betweenthe two images to measure similarity; the optimizer iterativelyadjusted the parameters of a transform to improve the cost func-tion, and the interpolator applied image transforms to calculatesub-voxel values. In this study, a multiresolution global

Fig. 1 (a) 3D rendered anatomical positions of the ascending aorta (AA) and pulmonary trunk(PT), shown in the anterior view (Top), with the medial aspect outlined as a dashed square), andin the posterior view (Bottom). (b) and (c) illustrate nonregistered images of the oldest adultpulmonary artery (PA) and AA (Top), and registered to their corresponding templates of young-est animal (Bottom). Note that the main side branch coming off of AA in ovine branches off tobrachiocephalic, left common carotid and left subclavian arteries at a more distal point.

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optimization method [22] with a correlation ratio cost functionwas used to determine the affine transformation variables of 3Drotation and scaling with the nearest neighbor interpolation,appropriate for binary data, used to apply the transform.

The AA registration was manually refined by aligning the seg-ments of coronary arteries proximal to sinuses of Valsalva and themain side branch with those of the template (Fig. 1(c)). To de-velop the PA centerline paths, the surface and local center pointswere aligned with the X3 axis (Fig. 2(a)), and the cylindrical polarcoordinates of local center points computed. The centerline pathof each PA was defined using fourth and second order polyno-mials to interpolate the radial and angle coordinates of local centerpoints as a function of the X3 coordinate, respectively. The 3Dcurvilinear feature of the centerline paths allowed refining PAs’registration and defining optimal scaling factors (Fig. 1(b)). TheAA and PA registered surface renderings were displayed togetherand each surface data set was re-cropped according to the afore-mentioned boundary definitions (Fig. 1(b) and 1(c)).

General Surface Fitting Considerations. Due to practicalconsiderations, the imaging data was acquired from animals at dif-ferent ages, so no direct physical marker information was available.Thus, for the present study we assumed that the relative circumfer-ential and longitudinal locations of surface points were locally pre-served throughout postnatal growth. This assumption was based onour observations that the overall shape of both arteries appeared to

be remarkably well preserved (see “Registration” results below).The 3D coordinates of arterial surfaces points were fit to an in-surface coordinate space using piece-wise cubic order polynomialfinite-element shape functions. The resulting surfaces were fit to acontiguous time based node fitting functions to estimate the effec-tive growth deformation field. Specific steps are given in thefollowing.

Surface Parameterization. A toroidal coordinate system wasdefined to allow use of the linear least-squares fitting method fordeveloping surface models of the PA and AA, where only a singlecoordinate was fitted [23], and resulted in a relatively uniform dis-tribution of data points in the in-surface coordinate plane [24].The local coordinate of each surface point was determined byfinding its orthogonal projection on the centerline path using thegradient-based unconstrained optimization routine of backtrackingline search, implemented in Mathcad (Parametric TechnologyCorporation. MA, USA).

Common Centerline Path Generation. In order to define allthe corresponding surfaces with respect to one coordinate system,a common centerline path was developed. The common center-line path of all PA surfaces was simply developed by interpolat-ing the radial and angle coordinates of individual centerline pathsas a function of X3 coordinate using second and fourth orderpolynomial functions, respectively, (Fig. 2(a)). The common

Fig. 2 (a) The common center axis of the PA along with individual center axes points (black) from multiple time points (scaled),with the Frenet frame and global Cartesian coordinate systems. Note the high degree of consistency between the centerlinepaths from the animals studied. Also shown in (b) are the AA and PA center axes from two perspectives to demonstrate largertortuosity of PA compared to AA. (c) A schematic surface parameterization and deformation showing the base vectors in the ref-erence to the end-diastolic loaded state of the youngest animal (15 kg) and those of current or ‘deformed’ geometry and the fit-ted centerline (CL).

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centerline path of AA surfaces was defined by applying the sameorder of polynomial interpolations to all the local center pointstogether.

Coordinate Transformation. Once the parameterization ofsurface points was complete, the local Cartesian coordinate wasdefined at each of the projection points on the centerline pathbased on the Frenet frame (Fig. 2(a)) [25]. This coordinate trans-formation is represented by

rðh; x3Þ ¼ xðh; x3Þ ¼ QðsÞ � x0ðh; sÞ þ yðx3Þ (1)

where s is the length along center axis path, x0 is the local coordi-nate of the surface point in the current configuration, with its pro-jected centerline path point y, and Q is the local 3D rotationmatrix (Appendix A).

Surface Fit Methodology. A finite element based surface fitwas developed to model the arterial surface geometry using 2D fi-nite element interpolation functions derived from the tensor prod-uct of 1D cubic Hermite element formulation to enforce C1

continuity [23]. The bicubic Hermite finite element has four func-tions defined at each node: the radial coordinate and its partial firstand cross derivatives with respect to local isoparametric coordi-nates n and g (which correspond to the h and s in Fig. 2). Thus,the radial coordinate of an internal surface point P is computed bysumming the product of 16 terms

qP ¼ wjki n; gð Þqjk

i (2)

where qjki is the vector of nodal variables, subscript i¼ 1,4 denotes

the node number, and subscripts j,k¼ 0,1 denote the order ofderivatives with respect to n and g, respectively.

Finite Element Grid Size. The very high density of datapoints from MR images greatly exceeded the minimum require-ment of 16 data points per element (with 16 nodal variables).However, the accurate reconstruction of the 3D surface in the to-roidal coordinate system through the finite element interpolationscheme required determination of the appropriate mesh resolutionfor capturing the prominent features while simultaneouslysmoothing the noise inherent in MR imaging and the segmentationprocesses. To this end, we developed the following approach todetermine the optimal number of elements required to accuratelyrepresent the endoluminal surface geometries. The mesh resolu-tion was determined based on the oldest PA geometry due to itsmore complex 3D shape compared to the AA. The in-planedimensions of the smallest feature of interest among all PA surfa-ces was resolved to be 6 mm and 7 mm in s and h directions,respectively. A finite element surface fit with relatively dense gridsize of 8 by 8 elements, that captured the features having twice ashigh of the spatial frequency as that of the smallest feature of in-terest, was initially generated. The surface fit was sampled atquarter size of the smallest targeted feature in h and s directions.A 2D FFT analysis was performed on the surface fit data usingMATLAB (The MathWorks Inc. MA, USA). The numbers of ele-ments in h and s directions were reduced iteratively until the 2DFFT power spectrum of the surface fit indicated significantsmoothing of the determined spatial noise.

2D Effective Surface Deformation. As stated earlier, we areconcerned with estimating the total surface growth in the loadedend-diastolic state, using an effective deformation gradient tensorFeff as the descriptor referenced to the youngest animal (of 15.3kg mass). We utilized a convective coordinate system under theassumption that deformations occur such that the coordinatelocations (i.e. h,s) for any surface point undergo minimal changes

during maturation (Fig. 2(c)). Starting with the endoluminal surfa-ces in the polar toroidal coordinate, base vectors were defined as

Ga ¼@R

@Ha ; ga ¼@r

@Ha (3)

where a¼ 1,2 and H1;H2� �

¼ h; sð Þ, Ga, and ga are the referen-tial and current base vectors (determined for each time point),respectively (Fig. 2(c)). Note that the base vectors are not neces-sarily orthogonal nor of unit magnitude [26]. The resulting straintensor was determined using [26]

eab ¼1

2gab � Gab� �

(4)

where gab ¼ ga � gb and Gab ¼ Ga �Gb. Since we are interested inan effective strain tensor expressed in current configuration, wedefined the eab as an effective Almansi strain tensor eab, so thatthe corresponding effective growth-induced stretches along thecoordinate lines were determined using

ka ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� 2eaa

gaa

r (5)

where the repeated indices do not imply summation. The measureof shear during growth was defined as

c ¼ cos�1 uh � usð Þ � cos�1 Uh � Us

� �(6)

where uh ¼ gh= ghj j, us ¼ gs= gsj j and Uh and Us are the corre-sponding unit base vectors in the reference configuration (here,again, defined as the youngest animal end-diastolic state). Since itwas shown that the mass of the animals highly correlated withtheir age and was known more exactly, mass was used to identifythe growth stage throughout the study [17]. The subsequentgrowth rates _ka (in units of kg–1) were obtained by computing theinstantaneous slope of the growth stretch ratio curves as a functionof mass, at each unit mass interval from 15 kg to 60 kg.

Time Interpolated Growth. To gain better insight into majortrends in spatiotemporal relationships, continuous surface defor-mation maps of the growing PA and AA were generated as fol-lows. The 3D surface shapes were approximated as a function oftime (again using the animal mass value) by interpolating thefour nodal surface fit variables. The radial coordinates werefound to follow a linear behavior, whereas the remaining nodalvariables were approximated using a second order polynomial.The scaling factors, obtained from registration, were interpolatedlinearly as a function of mass. The interpolated shape of each ar-tery at 15 kg was used as the referential geometry to obtain thespatiotemporal surface growth deformation map up to 60 kgusing Eqs. (1)–(5). This approach also had the advantage ofdeveloping an interpolated referential configuration that avoidedissues with individual specimen variations. Net results are pre-sented as mean 6 SD.

Results

Affine Registration. The scaled gross shapes of the PAs andAAs of the entire growth series closely matched with that of the15.3 kg template (Figs. 1(b) and 1(c)). The agreement of the cen-ter axis registration in particular provided additional evidence thatthe overall shape of the arteries were preserved during the growthperiod. Moreover, the mass and scaling factor values correlatedlinearly (r2¼ 0.85 with RMS error¼ 0.065 in PA and r2¼ 0.91with RMS error¼ 0.022 in AA), with every 10 kg increase inmass resulting in an increase of 0.2 in scale (unitless) in both the

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PA and AA size. The fourth and second order polynomial regres-sions of radial and angle coordinates very accurately approxi-mated the coordinates of combined PAs centerline paths, furthervalidating the accuracy of the registration method (RMSerror¼ 0.46 mm and 3.90 deg of radial and angle fits, respec-tively). The resulting AA common centerline path provided anaccurate representation of the combined local center coordinateswith angle coordinates of local center points staying relativelyconstant along the X3 coordinate (RMS error¼ 2.43 mm and 2.46deg for radial and angle fits, respectively). The PA and AA seg-ments of the centerline paths were calculated to be 24.01 mm and27.06 mm long, respectively.

Geometry. The results from the 2D FFT analysis on differentmesh densities indicated that the optimal number of equallyspaced element in h and s was a grid of 4� 3, respectively. Forthe sake of computational efficiency the data density was reducedby 50% to approximately 80 points per element. The 12 bicubicHermite element mesh provided an excellent representation ofthe endoluminal surface geometries (PA RMS error¼ 0.60 6 0.10mm, AA RMS error¼ 0.50 6 0.04 mm). The interpolatedarterial shapes with respect to animal mass provided a faithfulrepresentation of spatiotemporal geometric changes in boththe PA (RMS error¼ 0.80 6 0.033 mm for radial coordinates,RMS error¼ 0.28 6 0.11 mm and 1.61 6 0.22 for first derivativesin h and s directions, respectively, and RMS error¼ 1.706 0.88 mm for cross partial derivatives) and the AA (RMS

error¼ 0.64 6 0.034 mm for radial coordinates, RMSerror¼ 0.33 6 0.11 mm and 2.23 6 0.26 for first derivatives in hand s directions, respectively, and RMS error¼ 2.20 6 1.25 mmfor cross partial derivatives).

Overall Growth Patterns. With the improved approachesdeveloped herein compared to our previous study [17], we deter-mined the change in orientation of the cross-sections along eachartery relative to that of the STJ sections (i.e. the twist) using prin-cipal component analysis. We noted that that the orientation of thePA was largely maintained in the youngest (15 kg) and oldest ani-mals (60 kg), whereas the AA lost most of the observed twist(Fig. 3(a)). Interestingly, while it was again found that botharteries became increasingly tapered during postnatal develop-ment, the PA taper increased from 0.86 (i.e. nearly straight) to0.67 (highly tapered) while the AA taper increased only from 0.74to 0.62 (Fig. 3(b)). Another important difference between thearteries was the pronounced change in cross-sectional ellipticityin the PA with age, including increased changes along its length(Fig. 3(c)). In contrast, the AA exhibited both lower ellipticity(i.e. was more circular) and exhibited very little change with age(Fig. 3(d)).

Estimated Local Effective Surface Deformation Patterns.The effective growth shear strains were small both in the PA(c¼ 1.17 6 3.50 deg) and the AA (c¼ –0.37 6 1.40 deg). Thus,

Fig. 3 Basic geometric information of the growing AA and PA. (a) Orientation of cross-section major axes along the center axiscoordinate s (as an index of twist), showing while that of the PA remained mostly unaltered, the AA reduced with growth. (b)Alteration in arterial taper with growth, as measured by the ratio of cross-sectional areas of BFN or end-ROI to the STJ, was sub-stantially more prominent in the PA than AA. (c) Cross-sectional ellipticity of the PA increased substantially with growth (left),whereas in the AA it was relatively maintained (right). These simple measures underscore the geometric complexities of greatvessel growth during the post-natal period.

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the principal directions of growth deformation coincided closelywith the circumferential and longitudinal axes. This result indi-cates that the PA and AA grow almost exclusively along the cir-cumferential and longitudinal directions. Therefore, we report theeffective growth stretches only along the arterial axes. The follow-ing surface deformation results are based on interpolated surfacegeometries and stated differences in deformation values were allstatistically significant with p< 0.05.

Presenting the results for circumferential growth kh first, wenoted that most of the medial wall of the PA grew very little

circumferentially compared to other regions (kh¼ 1.12 6 0.012),while the corresponding segment of the AA medial wall was notas substantially inhibited (kh¼ 1.43 6 0.071) (Figs. 4 and 8(a) and8(b)). The circumferential growth of the PA lateral wall consider-ably decreased towards the BFN (from 1.69 6 0.11 to1.28 6 0.067, Figs. 5 and 8(b)). A similar, but less pronounced,trend was observed on the PA anterior wall, where towards theBFN, the kh values dropped from a mean of 1.70 6 0.066 to1.48 6 0.035 (Figs. 5 and 8(b)). The average circumferentialgrowth was significantly larger on the PA posterior wall compared

Fig. 4 Time-interpolated circumferential growth stretches on medial/posterior walls of PA (top)and AA (bottom). Both arteries demonstrated highly heterogeneous distributions, likely due tomutual mechanical interactions. Note in particular the effect of impingement on the PA medialwall by the AA, which resulted in reduced local deformations.

Fig. 5 Time-interpolated circumferential growth stretch on anterior and lateral walls of PA (top)and AA (bottom). Overall, this aspect of both vessels experienced more homogenous growth,with the PA of slightly higher heterogeneity.

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to the other three walls, and the pattern was relatively less hetero-geneous (1.73 6 0.056, Figs. 4 and 8(b)).

The largest circumferential growth in the AA occurred on alarge segment of its posterior wall with kh of 2.019 6 0.047 (Figs.4 and 8(b)). The kh growth patterns did not significantly changeon most of the lateral wall of the AA (1.66 6 0.056, Figs. 5 and8(b)). On the anterior wall of the AA, where the main side branchwas located, circumferential growth appreciably changed, with

the region located right after the side branch undergoing signifi-cantly smaller growth (1.28 6 0.029) compared to the rest of thataspect (1.47 6 0.067, Figs. 5 and 8(b)).

In contrast, the longitudinal growth kS was more homogenouscompared to the circumferential growth (Figs. 6 and 7). The pos-terior wall and to a lesser extent medial wall and a portion ofthe lateral wall of the PA showed larger longitudinal growth(2.50 6 0.10) compared to most of the anterior wall (1.99 6 0.094,

Fig. 6 Time-interpolated longitudinal growth stretch of posterior and a segment of medialwalls of PA (top) were significantly larger, due to a larger curvature, than that of the posteriorwall of AA (bottom)

Fig. 7 (a) Time-interpolated longitudinal growth stretch of anterior walls of PA (top) and AA(bottom), showing relatively uniformly changed with age with larger values in the PA at the 60kg growth stage due to its larger curvature than AA. (b) Longitudinal growth stretch profilesshown along four walls of PA (left) and AA (right)

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Figs. 6 and 7 and 8(c)). For most part, the longitudinal growth ofthe AA was relatively smaller and less heterogeneous with a meanof 1.80 6 0.023 (Figs. 6 and 7 and 8(c)).

As a means to represent effective regional growth, we deter-mined regional areal growth stretch as D¼ khkS, since the sheardeformations were small. Overall, the PA lateral wall consistentlyhad relatively smaller D, starting about 30% away from BFN(D¼ 2.77 6 0.18) compared to the rest of that wall(D¼ 3.72 6 0.13). The medial wall had on average the smallestgrowth (D¼ 2.88 6 0.28). The posterior wall grew the most(4.19 6 0.22) with anterior wall having the next largest growth(D¼ 3.31 6 0.26). The areal stretch of the AA was on averagesimilar in magnitude to those of the PA. The areal stretch of theAA exhibited only moderate heterogeneity (for the medial and an-terior walls D¼ 2.72 6 0.21 and 2.51 6 0.22, respectively). Thechange in surface area was larger on the lateral wall of the AA(D¼ 3.03 6 0.12) with the posterior wall growing the most(D¼ 3.44 6 0.29).

Growth Rates. The growth rates were computed in the STJ,middle regions (MRN) and the BFN region of the PA or end-ROIregion of the AA on all four aspects of each artery (Fig. 8(a),Table 1). Both circumferential and longitudinal growth ratesincreased with age as compared at four different growth stagesfrom youngest to oldest time points (except _kS of the AA was con-stant up to 35 kg growth stage). For the most part, _kh of the PAwas small up to about the 35 kg growth stage (about four monthsof age) and increased approximately linearly thereafter. However,the AA _kh was relatively more linear with smaller change ingrowth rates. The circumferential growth rate on the ‘contact’ seg-ment of the medial region of the PA was substantially smallerthan that of the corresponding segment of the AA medial wall.

Discussion

General Trends. In this first study of the spatial and temporalpostnatal growth deformation patterns of the AA and PA duringthe postnatal development, we determined that temporal growthpatterns were relatively consistent during the developmental

period. The spatial heterogeneity became progressively increasedwith growth, with the circumferential direction demonstrating alarger degree of spatial variation than longitudinal growth (Figs.4–8). The region of the medial PA wall in contact with AA hadthe smallest estimated circumferential growth at the 12-monthpostnatal (adult) stage. A large segment of the AA medial walland part of its posterior wall, which is proximal to the end-ROI(where the PA wraps around the AA), showed about 40–70%smaller circumferential growth. The lack of significant growth onthe sections of both arteries, where they are associated with eachother, implies that both mutually impose radial constraints. Thecircumferential growth rate on the AA medial wall in the PA-coupling region was about four times as fast as that of the corre-sponding region on the PA medial wall. Therefore, the constraintimposed by the AA on the PA had a substantially larger inhibitingeffect on the PA circumferential growth than vice versa.

The lateral wall and anterior walls of the PA showed, respec-tively, the next highest degrees of spatial heterogeneity. Theregion proximal to the BFN of the PA lateral wall, that coincideswith the anatomical position of the auricle of the left atrium, hadan average circumferential growth comparable to that of themedial wall of the PA and grew about a quarter as fast as the restof that wall. On the anterior wall of the PA, we speculate that thedecrease of at least 20% in circumferential growth stretch valuesin the vicinity of the BFN is due to the connections to Ligamen-tum Arteriosum in that region.

In a study by Huang et al. [27], increasing taper in the AA ofmice during the postnatal growth stage was reported. The afore-mentioned constraints proximal to the BFN on the lateral and an-terior walls of the PA probably have resulted in slowercircumferential growth compared to more distal regions, andexplain the increase in PA taper towards the BFN with age. Thepresence of the main side branch between the STJ and the end-ROI regions in the AA is the cause of its greater taper comparedto the PA (Figs. 1 and 3). The radial impingement of the PA onthe posterior wall of the AA proximal to the end-ROI limitsgrowth in that region and leads to an increase in axial taper as therest of the posterior wall of the artery is considerably less con-strained to grow.

Fig. 8 (a) Stretch profiles at the adult stage (60 kg) of the PA and AA taken from paths along four wall locations along thelength of each artery. (b) Circumferential growth stretch profiles of medial (M), lateral (L), anterior (A) and posterior (P) walls ofthe PA and AA. Note: In a small region after AA main side branch, towards which is oriented, slower circumferential growthrate was measured. (c) Corresponding longitudinal growth stretch profiles. Overall, these results underscored the pronouncedvariation in longitudinal growth stretch for the PA and circumferential growth stretch for both the AA and PA.

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The circumferential growth patterns of the rest of the AAappeared to be also guided by the presence of surrounding tissueimpingements. On anterior wall of the AA due to the slanted ori-entation of the main side branch towards the end-ROI, the regionlocated after the main branch showed an average of about 18%less growth than other areas of that wall (Figs. 5 and 8(b)), result-ing in the circumferential growth rate slowing down by a factor ofabout two and a half. Compared to the posterior wall, the rela-tively slower growth on the large axial region on the anterior wallextending to the lateral wall appears to be caused by the con-straints imposed by the superior vena cava and the left brachioce-phalic vessel.

The considerably larger kS of the PA posterior wall as com-pared to the anterior wall (Fig. 8(c)) was mainly due to its axialcurvature in presence of larger longitudinal than circumferentialsurface growth and essentially unchanged tortuosity of the center-line path during the whole growth period [17]. That is, in order tomaintain the axial curvature, the posterior PA wall had to growfaster than the anterior wall. This is consistent with a previousreport that the larger curvature is associated with larger growthdeformation [28]. The relatively smaller and significantly moreuniform longitudinal growth in the AA is due to the absence ofcomparable axial curvature to the PA (Fig. 2(b) and 8(c)). Theeven larger kS on the PA medial wall was a result of the concavedshape of the vessel in that region due to mechanical coupling withthe AA.

A Simplified Geometric Model Interpretation. In the presentwork, we relied on the evident geometric self-similarity of thearteries over the growth period. To more clearly demonstrate howthese observed geometrical phenomena must manifest themselves,we performed the following parametric study. A tapered cylindri-cal and two toroid phantoms with key dimensions taken from thePA measurements [29] were generated (Fig. 9). The toroidal for-mulae were modified to incorporate the mean ellipticity and tapermeasurements for each PA growth stage, with centerline path tor-tuosity maintained with growth (Appendix B). For the tapered cy-lindrical phantom, heterogeneous change in kh with growth isclearly evident (Fig. 8). In contrast, the presence of curvature in

the circular cross-section torus phantom produced heterogeneouskS patterns. When both effects were combined in the tapered ellip-tical toroid, which most closely represented PA geometry, theresulting phantom demonstrated both heterogeneous growth de-formation patterns as observed in the PA. Thus, any toroidal shellstructure with an ellipsoidal cross-section intrinsically requiresheterogeneous regional surface growth patterns to matchdelineated shape characteristics over the measured growth period(Fig. 8).

Physiological Functional Implications. Knowledge of arterialgrowth kinematics can shed light on mechanisms guiding devel-opment which can set the stage for the onset of a degenerative dis-ease [30]. As an example, it has been determined that duringnormal embryogenesis, the Truncus Arteriosus begins to split andform into the anterior PA and the posterior aorta [10]. Possiblydue to their common embryologic origin from a single outflowtract and their close association, there are disease conditions thatoriginate in one artery and eventually affect both arteries [31,32].This provided an additional reason to characterize the growth de-formation of both the AA and PA to quantify the effect of me-chanical connection of these two arteries.

The existence of and changes in geometric features such as cur-vature and cross-sectional torsion can also result in hemodynamicconditions leading to disease localization [33,34]. Additionally,due to the correlation between vessel geometry and its microstruc-ture, the change in morphology of the artery as it grows will resultin changes in its mechanical properties [35–37]. Furthermore, thedata on surface growth deformation patterns allows computationof changes in stress distribution, which is an important measure ofthe onset of disease and an indicator of predisposition to a varietyof vascular pathologies.

Finally, understanding normal arterial morphogenesis can yieldunique insight into the mechanisms of vascular adaptations andtheir important physiological factors. There is probably a dynamicinterplay between arterial growth kinematics, pulsatile blood flow,and arterial hemodynamics. The existence of taper in the AA hasbeen recognized as a physiological adaptation allowing for the opti-mization of pulsatile flow and favorable wave reflective properties

Table 1 The circumferential and longitudinal growth rates in PA and AA computed after about 35 kg growth stage in four circum-ferential and three longitudinal regions

Circumferential growth rate (reported as mean 6 SD in %/kg)

Circumferential region

Artery Axial region Medial Posterior Lateral Anterior Mean

PA STJ 1.5 6 0.5 2.3 6 0.7 2.4 6 1.3 1.9 6 0.6 2.0 6 0.8MRN 0.48 6 0.12 3.0 6 1.1 1.1 6 0.5 1.9 6 0.6 1.6 6 0.6BFN 0.24 6 0.03 1.9 6 0.6 0.48 6 0.13 0.83 6 0.16 0.9 6 0.2

AA STJ 1.6 6 0.06 2.7 6 0.7 1.3 6 0.2 1.5 6 0.5 1.8 6 0.4MRN 1.1 6 0.05 2.9 6 0.8 1.7 6 0.3 0.50 6 0.25 1.6 6 0.4

End-ROI 1.5 6 0.2 2.1 6 0.5 1.1 6 0.02 1.0 6 0.1 1.4 6 0.2

Longitudinal growth rate (reported as mean 6 SD in %/kg)

Circumferential region

Artery Axial region Medial Posterior Lateral Anterior

PA STJ 2.9 6 0.7 3.3 6 1.0 3.3 6 0.7 3.4 6 0.9MRN 4.6 6 1.1 4.2 6 0.9 3.6 6 0.9 2.3 6 0.6BFN 2.9 6 0.7 3.5 6 0.9 2.8 6 0.7 2.2 6 0.6Mean 3.5 6 0.8 3.7 6 0.9 3.2 6 0.8 2.6 6 0.7

AA STJ 1.7 6 0.4 2.5 6 0.7 1.5 6 0.7 1.7 6 0.4MRN 1.9 6 0.4 2.1 6 0.4 2.3 6 0.6 1.6 6 0.4

End-ROI 1.9 6 0.4 1.9 6 0.4 1.4 6 0.7 2.1 6 0.4Mean 1.8 6 0.4 2.2 6 0.5 1.7 6 0.7 1.8 6 0.4

Journal of Biomechanical Engineering JULY 2013, Vol. 135 / 071010-9

that are inherent to the design of the conduit arteries [38]. The arte-rial taper enables connecting segments of the cardiovascular systemthat have different wave propagation or input impedance propertiesin order to decrease blood flow fluctuations [39]. As reported byWells et al. [40] the viscoelasticity of the aortic wall decreases dur-ing postnatal development as measured by a drop in wave attenua-tion coefficient from juvenile to adult ovine. The increase in taper,therefore, may be compensating for this decline in arterial dampen-ing that would lead to a decrease in attenuation of the flow rate am-plitude. We also note that according to Melbin et al. [41], due tothe lower flexure rigidity relative to the in-plane stiffness of elasticarteries, ellipticity results in an initial deformation with constant pe-rimeter via pure bending at the onset of systole, resulting in analmost circular cross-sectional geometry.

Implications to Tissue Engineering and Cellular LevelGrowth Mechanisms. Overall, the geometric informationdetailed in this study is required to establish the baseline forachieving functional equivalence in engineered tissue replace-ments. Hoerstrup et al. [8], for example, reported the basic dimen-sional changes in the tissue engineered PA replacements. They

reported a maximum increase of 45% in length of the tissue-engineered ovine PA constructs within the 100-week postopera-tive period while our results indicated an increase of at least 200%in native PA length (kS¼ 2.0) within about a ten month postnatalmaturation period. They stated that mean internal diameterincreased by about 30% between 20 and 100 weeks; however,based on our results, the PA grew by about 40% in circumferenceafter 20 weeks (circumferential growth stretch: from 1.1 to 1.5).Clearly, growth metrics have not been yet fully achieved whencompared to native arteries. We furthermore maintain that theseand future results can guide the development of repair techniquesof both arteries in congenital abnormalities and restoring the nor-mal pulsatile arterial behavior.

Interestingly, relatively little is known about the exact regula-tory mechanisms underlying the biology of organ size determina-tion [42]. In one model of early postnatal life, the simultaneousupregulation of multiple specific growth-promoting genes resultsin periods of rapid growth [43]. Growth causes negative feedbackand subsequent down-regulation of these growth-promoting genes.As a result, growth slows, and an organism’s growth rate eventuallyapproaches zero. Finkielstrain et al. [44] concluded that there existsan extensive genetic program occurring during postnatal life.

Fig. 9 Geometric parametric studies using a tapered cylinder and two toroid phantoms, withkey dimensions taken from the PA measurements. For the tapered cylindrical phantom, hetero-geneous change in kh with growth is clearly evident. In contrast, the presence of curvature inthe circular cross-section toroid phantom produced heterogeneous ks patterns. When botheffects were combined in the tapered elliptical toroid, (which geometrically most closely repre-sented PA geometry), the resulting phantom demonstrated both heterogeneous growth defor-mation patterns as estimated in the PA in vivo.

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Alternatively, in the large conduit arteries, vascular cells arethought to transduce cyclic strain, caused by pulsatile blood flow,into signaling cascades which result in cell hypertrophy, hyperpla-sia, and extracellular matrix synthesis [45–48]. Although thesemechanisms of remodeling and growth have been proposed, thespecific signals that determine the rate of vascular remodeling (ei-ther in normal postnatal development or with congenital defects)have not yet been defined. Ultimately, finding links between molec-ular mechanisms of growth and the geometry of large vessels holdspromise for patients with congenital heart disease because hemody-namics are dictated by vascular geometry [49].

Limitations and Accuracy of the Current Approach. Sincethe animals used in the present work were sacrificed at varyingtimes for nonrelated studies, the same individual animals couldnot be imaged for the entirety of the study. However, we foundthat the growth trends in both arteries for all animals were consist-ent. We thus had to make the assumption that the relative spatiallocations with respect to in-surface coordinates were maintained(i.e. radially directed growth) within the growth period. Thisassumption was supported by the registration results (Figs. 1 and2), especially the similarity in shapes of the scaled centerlinepaths over the entire growth period (Fig. 2(a)), which demon-strated that the overall arterial shape was maintained. Thisincluded the maintenance of the orientations of the PA bifurcationand sinuses of Valsalva (Figs. 1(b) and 1(c)). We also noted thatthe circumferential location of AA-induced indentation on medialPA wall was preserved. This was evidenced using positionalmeasurements at three different cross-sectional locations alongthe PA surfaces, using the centerline path, over the growth period(Table 2).

Also relevant are the findings of our recent study wherein weperformed thickness, mechanical behavior, and detailed structuralanalyses on the PA arterial tissues [50]. Interestingly, we observedno changes in thickness or significant differences in the arterialwall circumferential stiffness from the juvenile to adult stages.This suggests that our arterial dimensions in the end-diastolic state[17] cannot be attributed to changes in elastic behavior, i.e., themedial wall did not become circumferentially stiffer with growth,so the smaller measured kh in that region is only attributable tolack of surface growth.

It should be noted here that our approach is not a model per se,but rather a means to estimate the effective surface growth as aform of deformation from a set of successive, co-registered 3D sur-face reconstructions. The only assumptions made with respect toour kinematic analysis is that the arterial surfaces only grow in-plane, and that the relative position of material points in the scaled(h,s) coordinate remains constant.

Interestingly, our parametric studies using simulated toroidalshapes (Fig. 9) suggest that our initial results are a reasonable first-approximation to the actual effective growth patterns. For example,the longitudinal effective growth stretch ks on the inner curvaturewas about 1.29 in both the PA (Fig. 6, top-right image) and the

toroidal phantom with ellipsoidal cross-section (Fig. 8, bottomright image). Similarly, kh at the STJ was found to vary from 1.3 to1.8 for the oldest animal (Fig. 4, top right image) and 1.54 to 1.74for the toroidal phantom (Fig. 9, center bottom image).

Summary. Overall, our results indicated that the spatial andtemporal surface growth deformation patterns of both arterieswere heterogeneous, including an increase in taper in both arteriesand increase in cross-sectional ellipticity of the PA. Contactbetween the PA and AA resulted in increasing spatial heterogene-ity in postnatal growth, with the PA demonstrating the greatestchanges. Parametric studies using simplified geometric models ofcurved arteries during growth suggest that heterogeneous effectivesurface growth deformations must occur to account for thechanges in measured arterial shapes during the postnatal growthperiod. Results of this study clearly underscore and extend ourearlier observations that functional growth of the PA and AA dur-ing postnatal maturation involves complex geometric adaptations.Moreover, they help to lay the basis for functional duplication forrepair and replacement by defining critical geometric metrics.How this process is biologically regulated remains a major ques-tion for both native and engineered tissue PA replacements.

Acknowledgment

The National Institutes of Health Grant No. R01 HL089750supported this work. The authors are grateful to Dr. Robert Tam-buro for his technical assistance with the image registrationproject.

Appendix A: Coordinate Transformation

The local coordinate of the surface point, x0, with its projectedcenterline path point, y, and Q are the 3D rotation matrix definedby local Frenet frame as

x0 ¼q h; sð Þcos hð Þq h; sð Þsin hð Þ

0

264

375

y ¼R x3ð Þcos H x3ð Þð ÞR x3ð Þsin H x3ð Þð Þ

x3

264

375

Q sð Þ ¼n1 sð Þ b1 sð Þ t1 sð Þn2 sð Þ b2 sð Þ t2 sð Þn3 sð Þ b3 sð Þ t3 sð Þ

264

375

(A1)

where q is the radial coordinate defined as a function of circum-ferential location, h, and length along the centerline path given by

s x3ð Þ ¼ðx3

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidy1

dx3

� �2

þ dy2

dx3

� �2

þ1

s� dx3 (A2)

Appendix B: Modified Torus

Equation of torus in cylindrical polar coordinates is given by

X1 ¼ rmaj cos hð ÞX2 ¼ rmin sin hð Þ þ Rð Þ cos /ð ÞX3 ¼ rmin sin hð Þ þ Rð Þ sin /ð Þ

(B1)

where h and / define circumferential and centerline path coordi-nate; R is the radius of the torus centerline and

rmin /ð Þ ¼ rSTJ þffiffiffiffiffiffiffiffiffiffitaperp

� 1ð Þ � rSTJ �R � /

L(B2)

Table 2 Location of the AA induced indentation on medial PAwall with growth as measured in three different cross-sectionsof PA surfaces along centerline path at 15 kg, 30 kg, 40 kg,50 kg, and 60 kg time points. The reported angular locationvalues were taken at the center of the span and were accurateto within approximately 63 deg and varied in circumferentialposition by the same amount. Interestingly, they did notundergo changes with growth.

Distance of cross-sectionsfrom STJ, s

Location h(deg)

Angular span(deg)

30% 175 3860% 167 3790% 145 25

Journal of Biomechanical Engineering JULY 2013, Vol. 135 / 071010-11

where L is the length of centerline, rmaj ¼ ecc� rmin with eccbeing the mean cross-sectional ellipticity.

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